Ups and Downs Notation
Ups and Downs (or ^v) is a notation system developed by Kite that can notate almost every EDO. The up symbol "^" and the down symbol "v" indicate raising/lowering a note (or widening/narrowing an interval) by one EDOstep. The mid symbol, "~" is for intervals exactly midway between major and minor, e.g. 3\24 is a mid 2nd. The mid 4th (~4) is midway between perfect and augmented, i.e. halfway-augmented, and the mid 5th (~5) is a halfway-diminished 5th.
Ups and downs can also notate any rank-2 temperament, although some temperaments require an additional pair of accidentals, lifts and drops (/ and \). In this context, an up or a lift represents sharpening by a comma that has been tempered, but not tempered out. For example, in Triyo aka Porcupine, an up/down represents raising/lowering by a tempered 81/80, and lifts/drops aren't used. In practice, the two uses of the notation often coincide perfectly. Triyo is supported by both 15-edo and 22-edo, and both EDOs map 81/80 to one EDOstep. Thus if Triyo is tuned to 15-edo, an up simultaneously means both a tempered 81/80 and 1\15. Likewise, if tuned to 22-edo, the up means both 81/80 and 1\22. If not tuned to an EDO at all, then the up only means 81/80. Thus a piece written in Triyo can be converted to a piece written in 22-edo by simply writing "22-edo" on the top of the page.
This page only discusses notation of EDOs. For notation of rank-2 and rank-3 temperaments, see the pergens article. For more on EDO notation, see the Notation guide for edos 5-72, which also covers chord names, slash chords, staff notation, key signatures, and scale trees.
Explanation -- a 22-edo example
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-edo is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDOstep. C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-edo as long as you remember that C# and Db are different notes.
In contrast, 22-edo is hard to notate because 7 fifths are three EDOsteps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the 5/4, usually a major 3rd, becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!
The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol to mean "sharpened by one EDOstep". 22-edo can be written C - Db - ^Db - vD - D - Eb - ^Eb - vE - E - F etc. The notes are pronounced up-D-flat, down-D, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. The up or down comes before the note name to make naming chords easy.
The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - ^C - vC# - C# - D - ^D - vD# - D# - E etc.
The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.
Relative notation and interval arithmetic
Ups and downs can be used not only for absolute notation (note names) but also for relative notation (intervals, chords and scales). Relative notation for 22-edo intervals: P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^4/d5 - vA4/^d5 - A4/v5 - P5 etc. That's pronounced upminor 2nd, downmajor 3rd, etc. You can apply this pattern to any 22-edo key. The notes without ups or downs always form a chain of fifths.
A core principle of ups and downs notation is that interval arithmetic is always preserved. Ups and downs are simply added in:
|conventional||with ups and downs||(cancelling)||(combining)|
|interval between two notes||C to E = M3||^C to E = vM3||C to ^E = ^M3||^C to ^E = M3||^C to vE = vvM3|
|note plus an interval||C + M3 = E||^C + M3 = ^E||C + ^M3 = ^E||^C + vM3 = E||^C + ^M3 = ^^E|
|sum of two intervals||M2 + M2 = M3||^M2 + M2 = ^M3||M2 + vM2 = vM3||^M2 + vM2 = M3||vM2 + vM2 = vvM3|
Conventionally, in C you use D# instead of Eb when you have a Gaug chord. You have the freedom to spell your notes how you like, to make your chords look right. Likewise, in 22-edo, Db can be spelled ^C or vB# or even ^^B (double-up B).
From the pergens article: "Conventional notation is generated by the octave and the 5th, and the notation (not the tuning itself) is rank-2. Each additional pair of accidentals increases the notation's rank by one, analogous to adding primes to a JI subgroup. Enharmonic intervals are like commas in that each one reduces the notation's rank by one (assuming they are linearly independent). Obviously, the notation's rank must match the actual tuning's rank. Therefore the minimum number of enharmonics needed always equals the difference between the notation's rank and the tuning's rank."
Since 22edo is rank-1, and conventional notation plus ups and downs is rank-3, two enharmonic intervals are needed to define the notation: v3A1 and vm2. Either interval can be added to or subtracted from any note to respell the note. For example, ^C + vm2 = Db and ^^Eb + v3A1 = vE. Any combination of these two enharmonic intervals is also an enharmonic interval, for example their sum v4M2.
For staff notation, put an up or down to the left of the note and any sharp or flat it might have. Like sharps and flats, an up or down applies to any similar note that follows in the measure. If F is upped, any other F in the same octave inherits the up, but an F# doesn't. Key signatures follow the conventional practice, expanded to allow for double-sharps and double flats in some EDOs. For example, 19-edo has the key of Bbb with a key signature of Bbb Ebb Ab Db Gb Cb Fb. Some EDOs have upped/downed tonics, e.g. 24-edo has the key of vD with a key signature of F# C# (v). The (v) is a "global down" that downs all 7 notes of the vD scale. For more on staff notation, see the Notation Guide for EDOs 5-72.
Placement of the up or down
It might seem more natural to place the up after the note, for example B^ or Bb^. But the up must come first, to make chord names unambiguous. B^m could mean either a minor chord rooted on B^ or an upminor chord rooted on B. (Chord names are explained fully below.)
The issue arises because while English normally places the adjective before the noun, it doesn't do so with sharps and flats. A flattened B should logically be called "flat B" not "B flat", and be written bB not Bb. If it were, then it would seem very natural to have the up come first, as in ^bB. This would be the typical English adjective-adjective-noun construction. Instead we must use ^Bb, an unnatural adjective-noun-adjective construction. This issue fortunately arises only for note names. On the staff, the flat comes before the note, so naturally the up comes before the flat. In relative notation, the quality comes before the interval, as in minor 3rd and augmented 4th, or in jazz terms flat 3rd and sharp 4th. So terms like upminor 3rd and downsharp 4th have a natural adjective-adjective-noun construction.
Examples: EDOs 12-24
Sharp-1, flat-2, etc. refer to the number of EDOsteps made by seven 5ths minus four 8ves. All sharp-1 and flat-1 edos can be notated without ups and downs, because the up is exactly equivalent to a sharp or flat.
A ring is a circle of 5ths. In multi-ring (aka ringy) edos like 14, 15 and 24, a single ring doesn't contain all the edo's notes. In contrast, edos like 12, 19 and 22 are single-ring. It's possible to notate any single-ring edo with conventional notation if notes are permitted to be out of order (e.g. 22edo could have C Db B# C# D). But multi-ring edos absolutely require ups and downs.
13-edo and 18-edo aren't compatible with heptatonic notation, because the minor 2nd is descending. Thus the minor 3rd is flatter than the major 2nd, the 4th is flatter than the major 3rd, etc. These edos are best notated using the 2nd best fifth, i.e. as 13b and 18b.
12-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: d2.
There are two ways to notate 13b-edo: with sharp lowering the pitch, and major/aug narrower than minor/dim, or with sharp raising the pitch, and major/aug wider than minor/dim. The enharmonic intervals for the former notation are ^3A1 and vM2. For the latter they are v3A1 and vm2.
|sharp lowers the pitch,
major narrower than minor
|sharp raises the pitch,
major wider than minor
Because every 14-edo interval is perfect, the quality can be omitted. Sharps and flats can also be omitted. 14-edo contains 2 rings of 7-edo: an up/down-ring and a plain-ring. Enharmonic intervals: A1 and vvm2.
15-edo contains 3 rings of 5-edo: an up-ring, a down-ring, and a plain-ring. Enharmonic intervals: v3A1 and m2.
16-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either AA2 or dd2.
|sharp lowers the pitch,
major narrower than minor
|sharp raises the pitch,
major wider than minor
17-edo is sharp-2 and thus has mid intervals. Enharmonic intervals: vvA1 and vm2.
18b-edo contains 2 rings of 9-edo: an up/down-ring and a plain-ring. There are two ways to notate it. Enharmonic intervals: either ^^A1 and vvM2, or vvA1 and vvm2.
major is narrower
major is wider
19-edo is sharp-1, thus doesn't need ups and downs. Enharmonic interval: dd2.
20-edo contains 4 rings of 5-edo: an up-ring, a down-ring, a double-up/down-ring, and a plain-ring. Enharmonic intervals: v4A1 and m2.
Because every 21-edo interval is perfect, the quality can be omitted. 21-edo contains 3 rings of 7-edo: an up-ring, a down-ring and a plain-ring. Enharmonic intervals: A1 and v3m2.
22-edo is sharp-3. Enharmonic intervals: v3A1 and vm2.
23-edo is flat-1, thus doesn't need ups and downs. There are two ways to notate it. Enharmonic interval: either A32 or d32.
major is narrower
major is wider
24-edo contains 2 rings of 12-edo: an up/down-ring and a plain-ring. Enharmonic intervals: vvA1 and d2.
Extremely Large EDOs
In theory, every edo can be notated with ups and downs only. For example, in 159-edo, 11/8 above C would be ^7F. But large exponents can be avoided if the edo is multi-ring (if the circle of 5ths doesn't include every note). Ups and downs are used within a ring, and lifts and drops (/ and \, see the pergen page) are used to label each ring. 159-edo has 3 rings of 53-edo. Many people who work with 159-edo are familiar with 53-edo, and can read the lifts and drops as small inflections of the familiar 53-edo notation. 11/8 becomes ^^/F. See the 159edo notation page for the complete notation.
Chords and Chord Progressions
Chord names are based on jazz chord names. See Jim Aiken's book A Player's Guide to Chords & Harmony. Alterations are enclosed in parentheses, additions never are.
In perfect EDOs (7, 14, 21, 28 and 35), every interval is perfect, and there is no major or minor. In the following list of chord names, omit major, minor, dim and aug. Substitute up for upmajor and upminor, and down for downmajor and downminor. The C-E-G chord is called "C perfect" or simply "C".
An up or down between the chord root and the chord type (e.g. C^m7) raises or lowers the 3rd, and also the 6th, 7th or 11th, if present. Thus C down-nine is the usual C9 chord with the 3rd and 7th downed: Cv9 = C vE G vBb D. A mid-something chord has a mid 3rd, 6th, 7th, and/or 11th. Mnemonic: every other note of a stacked-3rds chord with a 6th below the root is affected: 6th - root - 3rd - 5th - 7th - 9th - 11th - 13th.
The rationale for this rule is that a chord often has a note a perfect fourth or fifth above the 3rd. Furthermore, in many EDOs, upfifths, downfifths, upfourths and downfourths will all be quite dissonant and rarely used in chords. Thus if the 3rd is upped or downed, the 6th or 7th likely would be too. However the 9th likely wouldn't, because that would create an upfifth or a downfifth with the 5th. By the same logic, if the 7th is upped or downed, the 11th would be too.
Every conventional chord can accept such an up or down, with one exception: it's pointless to down a C5 chord, because there is no 3rd, 6th or 7th to alter. Thus Cv5 is invalid, and "C down-5" means C(v5) = C E vG.
Chord progressions use ups/downs notation to name the roots, e.g. Cv - Gv - vA^m - F or Iv - Vv - vVI^m - IVv. In relative notation, never use lower case roman numerals for minor chords, because both vIIm and VIIm would be written vii.
The major chord and various alterations of it:
- C E G = C = "C" or "C major" (in perfect EDOs, "C" or "C perfect")
- C ^E G = C^ = "C up" or "C upmajor"
- C vE G = Cv = "C down" or "C downmajor" (in EDOs 10, 17, 24, 31, etc., C~ = "C mid")
- C vvE G = Cvv = "C double-down" (in EDOs 20, 27, 34, 41, etc., C~ = "C mid", in EDOs 25, 32, 39, 46, etc. C^~ = "C upmid")
This table shows how altering the 3rd or the 5th affects the name of the triad The conventional abbreviations for aug and dim are + and o. These are rather cryptic, and can be replaced with the more obvious and intuitive a and d. Likewise the symbols Δ and − can be replaced with M and m.
|what's downed||C E G||C Eb G||C F G||C D G||C E G#||C Eb Gb|
Many EDOs have notes between the major 3rd and the perfect 4th, creating triads impossible in 12-edo, such as:
- C Fb G = C(d4) or C(b4) = "C dim-four" or "C sus-flat-four"
- C E# G = C(A3) or C(#3) = "C aug-three" or "C sus-sharp-three"
- C Ebb G = C(d3) or C(bb3) = "C dim-three" or "C sus-double-flat-three"
- C D# G = C(A2) or C(#2) = "C aug-two" or "C sus-sharp-two"
The "sus" is needed so that C(#2) doesn't sound like C#2, which is C# D# G#.
Sixth and seventh chords:
If the 7th is not a perfect 5th or a dim 5th above the 3rd, the chord is named as a triad with an added 7th.
- C E G Bb = C7 = "C seven"
- C vE G Bb = Cv,7 = "C down add-seven"
- C E G vBb = C,v7 = "C add down-seven"
- C vE G vBb = Cv7 = "C down seven"
All 7th chords follow this same pattern. Likewise, if a 6th is not a P4 or A4 above the 3rd, it's an "add-6" chord. Permitting add-7 chords has the added benefit that the wordy "minor-7 flat-5" and the illogical "half-dim" can be replaced with "dim add-7", written Cd,7.
In the table below, if a chord is bolded, the comma must be spoken as "add". In non-bolded names, the comma can be omitted when writing the chord, although that might make the chord name less readable.
|maj7||dom7||min7||min7(b5) or half-dim or dim-add-7||dim7||maj6||min6|
|what's downed||C E G B||C E G Bb||C Eb G Bb||C Eb Gb Bb||C Eb Gb Bbb||C E G A||C Eb G A|
|3rd, 5th, 7th||CvM7(v5)||Cv7(v5)||Cvm7(v5)||Cvm7(vb5)||Cvø(v5)||Cvd(v5)v7||Cvd7(v5)||Cvo7(v5)||Cv6(v5)||Cvm6(v5)|
Various unusual tetrads:
- C vE G ^Bb = Cv,^7 = "C down up-seven" (in EDOs 17, 24, 31, etc. C~7 = "C mid-seven")
- C E G A# = C,#6 or C,A6 = "C add sharp-six" or "C add aug-six"
- C E G Ab = C,b6 or C,m6 = "C add flat-six" or "C add minor-six"
- C E G Bbb = C,d7 or C,bb7 = "C add dim-seven" or "C add double-flat-seven" (19-edo's 4:5:6:7 chord)
- C E G B# is C,#7 or C,A7 = "C add sharp-seven" or "C add aug-seven"
- C E G Cb = C,b8 or C,d8 = "C add flat-eight" or "C add dim-eight"
In bolded chords, the comma is spoken as "add". Double alterations need only a single pair of parentheses, e.g. C vE vG B D is named CM9(v3v5). Double additions mostly need only a single comma, e.g. C E G vBb vD is named C,v7v9. But certain 6/9 chords require two commas. In these chords, marked with an asterisk *, only the first comma is spoken as "add".
|what's downed||C E G D||C E G B D||C E G Bb D||C Eb G Bb D||C E G Bb Db||C E G A D||C Eb G A D|
|3rd||Cv,9||CM9(v3)||C9(v3)||Cm9(v3)||Cv,7b9||Cv,6,9 *||Cvm,6,9 *|
|3rd, 9th||Cv,v9||Cv,M7v9 or
|3rd, 5th, 6th/7th||------||CvM9(v5)||Cv9(v5)||Cvm9(v5)||Cv7(v5)b9||Cv6(v5)9||Cvm6(v5)9|
|3rd, 5th, 9th||Cv(v5)v9||Cv(v5)M7v9 or
|3rd, 6th/7th, 9th||------||CvM7,v9||Cv7,v9||Cvm7,v9||Cv7,vb9||Cv6,v9||Cvm6,v9|
|5th, 6th/7th, 9th||------||C(v5)vM7v9||C(v5)v7v9||Cm(v5)v7v9||C(v5)v7vb9||C(v5)v6v9||Cm(v5)v6v9|
|3rd, 5th, 6th/7th, 9th||------||CvM7(v5)v9||Cv7(v5)v9||Cvm7(v5)v9||Cv7(v5)vb9||Cv6(v5)v9||Cvm6(v5)v9|
In 22-edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19-edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because "major 3rd" is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22-edo major chords sound ru (7-under) and 19-edo major chords sound yo (5-over).
A chord quality like "major" refers not to the sound but to the function of the chord. If you want to play a I - VIm - IIm - V - I progression without pitch shifts or tonic drift, you can do that in any EDO, as long as you use only major and minor chords. The notation tells you what kind of chord can be used to play that progression. In 22-edo, the chord that you need sounds like a ru chord.
In other words, I - VIm - IIm - V - I in just intonation implies Iy - VIg - IIg - Vy - Iy, but this implication only holds in those EDOs in which major sounds yo. Because 22-edo's yo chord 0-7-13 = 0¢-382¢-709¢ is downmajor, it doesn't work in that progression.
Another example: I7 - bVII7 - IV7 - I7. To play this progression without shifts or drifts, the 7th in the I7 chord must be a minor 7th. in 22-edo, that 7th sounds zo (7-over). In 19-edo, it sounds gu (5-under).
Ups and downs solfege
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
The initial consonant remains as before: D, R, M, F, S, L and T
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
The vowels are pronounced as in Spanish or Italian, and the pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
The 2nd vowel is optionally separated from the first by an "h", a "w", or a "y"
Thus vC# is Deo, pronounced as Deo or Deho or Dewo or Deyo.
This suffices for many but not all EDOs, as some require triple sharps or quadruple ups.
Fixed-do solfege examples:
- Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
- Da = C, Dae = ^C, Dai = ^^C, Dao = vC, Dau = vvC, Daa = ^^^C
- De = C#, Dee = ^C#, Dei = ^^C#, Deo = vC#, Deu = vvC#, Dea = vvvC#
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
- Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
- Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1
Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim
- Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2
- Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2